It's probably about reducing maintenance costs. The public API is implemented by software which calls internal Netflix services/APIs. Over time, those services accrue technical debt and need to be refactored or rewritten. Every system that uses those internal services needs to be updated to match.
A published API makes this more difficult. The behavior of published APIs, particularly those that are in wide use, can not change in any way without raising the ire of developers. This restricts the internal refactoring you can do and makes the whole exercise (already complicated enough) even more challenging.
Working with a small, curated list of apps means that they can coordinate with developers on API changes rather than worrying about maintaining compatibility. This lowers their maintenance costs and increases their architectural agility.
A published API makes this more difficult. The behavior of published APIs, particularly those that are in wide use, can not change in any way without raising the ire of developers. This restricts the internal refactoring you can do and makes the whole exercise (already complicated enough) even more challenging.
Working with a small, curated list of apps means that they can coordinate with developers on API changes rather than worrying about maintaining compatibility. This lowers their maintenance costs and increases their architectural agility.